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Question Number 139771 by Engr_Jidda last updated on 01/May/21

Commented by mohammad17 last updated on 01/May/21

$$yessirrealyimsorycanyousolvethis?$$

Commented by mr W last updated on 01/May/21

$$takecare!$$$$\sqrt[\frac{1}{2}]{3}=3^2\ne 3^{\frac{1}{2}}$$

Commented by mohammad17 last updated on 01/May/21

$$sirif\sqrt[\frac{1}{2}]{3}=3^{2}then\sqrt[\frac{1}{7}]{3}=3^{7}itsrightorno$$

Commented by mr W last updated on 01/May/21

$$yes.{it}^{\prime }sright.$$

Commented by mohammad17 last updated on 01/May/21

$$thankyouverymuchsir$$

Answered by mr W last updated on 01/May/21

$$\sqrt[x]{a}=a^{\frac{1}{x}}$$$$\Rightarrow \sqrt[\frac{1}{x}]{a}=a^{\frac{1}{\left(\frac{1}{x}\right)}}=a^x$$$$$$$$\sqrt[\frac{1}{x}]{{\left(\sqrt[\frac{1}{x}]{x^x}\right)}^x}=4$$$$\sqrt[\frac{1}{x}]{{\left(x^{x^2}\right)}^x}=4$$$${\left(x^{x^2}\right)}^{x^2}=4$$$$x^{x^2{x}^2}=4$$$$x^{x^4}=4$$$$\Rightarrow x^4=4$$$$\Rightarrow x=\pm \sqrt[4]{4}=\pm \sqrt{2}$$

Commented by Ankushkumarparcha last updated on 01/May/21

$$ifuusepowertowerthenthisequationhasnosolution$$$$\because rangeR\in [\frac{1}{e},e]$$

Commented by mr W last updated on 01/May/21

$$i{didn}^{\prime }tusepowertower.$$$$theoriginalequationisequivalentto$$$$x^{x^4}=4.$$$$andeqn.x^{x^4}=ahasalwayssolutionif$$$$a⩾\approx 0.9121$$

Answered by Ankushkumarparcha last updated on 01/May/21

$$Solution:\sqrt[\frac{1}{x}]{{\left(\sqrt[\frac{1}{x}]{x^x}\right)}^x}=4=>x^{x^4}=4(\because \sqrt[\frac{1}{a}]{b}=b^a)$$$$Byobservingweget,x=\sqrt{2}$$$$$$

Commented by mr W last updated on 01/May/21

$$x=-\sqrt{2}isalsosolution.$$

Answered by Ar Brandon last updated on 01/May/21

$${\mathrm{x}}^{{\mathrm{x}}^4}=4\Rightarrow {\mathrm{x}}^4\mathrm{lnx}=\ln 4$$$$\Rightarrow {4\mathrm{x}}^4\mathrm{lnx}=4\ln 4$$$$\Rightarrow 4\mathrm{lnx}\cdot {\mathrm{e}}^{4\mathrm{lnx}}=\ln 4\cdot {\mathrm{e}}^{\ln 4}$$$$\Rightarrow {\mathrm{W}}_0\left(4\mathrm{lnx}\right)={\mathrm{W}}_0\left(\ln 4\right)$$$$\Rightarrow 4\mathrm{lnx}=\ln 4\Rightarrow {\mathrm{x}}^4=4\Rightarrow \mathrm{x}=\pm \sqrt{2}$$

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